Cotangent is Reciprocal of Tangent

Theorem

Let $\theta$ be an angle such that $\cos \theta \ne 0$ and $\sin \theta \ne 0$.

Then:

$\cot \theta = \dfrac 1 {\tan \theta}$

where $\tan$ and $\cot$ mean tangent and cotangent respectively.

Let a point $P = \tuple {x, y}$ be placed in a cartesian plane with origin $O$ such that $OP$ forms an angle $\theta$ with the $x$-axis.

Then:

$\ds \cot \theta$

$=$

$\ds \frac x y$

$\ds $

$=$

$\ds \frac 1 {y / x}$

$\ds $

$=$

$\ds \frac 1 {\tan \theta}$

$\tan \theta$ is not defined when $\cos \theta = 0$, and $\cot \theta$ is not defined when $\sin \theta = 0$.

$\blacksquare$